This is always the first question people would ask when planning about cycling tour around Taiwan. BUT, there is no standard or correct answer to it, mostly depending on what you want to see and when you plan to set off. Here are the some thoughts about both directions which you can take into consideration before planning or making the final decision.
Wind is blowing in difference direction according to seasons. During summer time, mostly wind is coming from south-west direction, which make travelling from north to south a bit challenge comparing to other direction. During winter time, headwind is against you when you are riding the bike from south to north, and make it very difficult to move forward. However, since you are traveling all around Taiwan in the same season, so the only difference is you choose to have the sweet and easy rides first or save the best to the last. Most people would suggest to go with clockwise in summer time and counter-clockwise in winter time, so you have some time to prepare yourself before meeting the wind challenge. (Believe or not, wind is way tougher than hills for cyclists!)
Another point to consider is the scenic views on your journey. Biking clockwise in Taiwan means you are always riding on the opposite side of the coastlines. It is safer in terms of riding, but when you wish to get closer to the sea and take the wonderful view of Pacific in the east coast, you would have to cross the road and get to the other side in order to do this. And frequent enough, you will find this annoying. Riding your bicycle around Taiwan in counter-clockwise direction, meaning you will always have the benefit of staying close to the sea, but that means more dangerous when it is only a cliff right next to you when riding in some parts of Taitung and Hualien.
Whatever your decision may be, keep enough time for yourself during the trip and just enjoy the ride in Taiwan!